Abstract
We consider bivariate polynomials over the skew field of quaternions, where the indeterminates commute with all coefficients and with each other. We analyze existence of univariate factorizations, that is, factorizations with univariate linear factors. A necessary condition for existence of univariate factorizations is factorization of the norm polynomial into a product of univariate polynomials. This condition is, however, not sufficient. Our central result states that univariate factorizations exist after multiplication with a suitable univariate real polynomial as long as the necessary factorization condition is fulfilled. We present an algorithm for computing this real polynomial and a corresponding univariate factorization. If a univariate factorization of the original polynomial exists, a suitable input of the algorithm produces a constant multiplication factor, thus giving an a posteriori condition for existence of univariate factorizations. Some factorizations obtained in this way are of interest in mechanism science. We present an example of a curious closed-loop mechanism with eight revolute joints.
Highlights
By (H, +, ·) we denote the skew field of real quaternions with the usual addition and non-commutative multiplication
A fundamental theorem of algebra holds true for polynomials in H[t]: Each non-constant univariate quaternionic polynomial admits a factorization with linear factors (c.f. [3,4,15])
The most promising article in this context is [16] by Skopenkov and Krasauskas. They introduce a technique for the factorization of bivariate quaternionic polynomials of bi-degree (n, 1), where n ∈ N0 is an arbitrary non-negative integer
Summary
The norm polynomial of a univariate quaternionic polynomial of degree greater than one is a product of at least two irreducible real polynomials that play an important role in the computation of factorizations This is in contrast to the bivariate case where factorizability of the norm polynomial is exceptional and not a sufficient condition for existence of factorizations. The most promising article in this context is [16] by Skopenkov and Krasauskas They introduce a technique for the factorization of bivariate quaternionic polynomials of bi-degree (n, 1), where n ∈ N0 is an arbitrary non-negative integer. In the present paper we show that similar identities hold for the Beauregard polynomial (1) but are generally true: For any polynomial Q ∈ H[t, s] satisfying the necessary factorization condition there exists a real polynomial K ∈ R[t] (or K ∈ R[s]) such that KQ admits a univariate factorization. 2. multiplication with a real polynomial factor is admissible as it does not change the underlying rational motion
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